Two bodies with masses `M_(1)` and `M_(2)` are initially at rest and a distance `R` apart. Then they move directly towards one another under the influence of their mutual gravitational attraction. What is the ratio of the distances travelled by `M_(1)` to the distance travelled by `M_(2)`?

Options

A. `(M_(1))/(M_(2))`
B. `(M_(2))/(M_(1))`
C. `1`
D. `1/2`

Explanation

To find the ratio of the distances traveled by two bodies with masses \( M_1 \) and \( M_2 \) as they move towards each other under mutual gravitational attraction, we can follow these steps:

### Step 1: Understand the System
We have two masses \( M_1 \) and \( M_2 \) separated by a distance \( R \). Initially, both masses are at rest. As they move towards each other due to gravitational attraction, we need to find the distances they travel until they meet.

### Step 2: Define the Center of Mass
The center of mass (CM) of the system can be calculated using the formula:
\[ \text{CM} = \frac{M_1 \cdot x_1 + M_2 \cdot x_2}{M_1 + M_2} \]
where \( x_1 \) is the distance traveled by \( M_1 \) and \( x_2 \) is the distance traveled by \( M_2 \).

### Step 3: Set Up the Distance Relationship
Since both bodies move towards each other and no external forces are acting on them, the center of mass will remain stationary. Thus, the distances traveled by the two masses can be expressed in terms of their masses:
\[ M_1 \cdot x_1 = M_2 \cdot x_2 \]

### Step 4: Express Distances in Terms of Total Distance
The total distance \( R \) that separates the two bodies is equal to the sum of the distances they travel:
\[ x_1 + x_2 = R \]

### Step 5: Solve for the Ratio
From the previous equations, we can express \( x_1 \) in terms of \( x_2 \):
\[ x_1 = \frac{M_2}{M_1} x_2 \]
Substituting this into the total distance equation gives:
\[ \frac{M_2}{M_1} x_2 + x_2 = R \]
Factoring out \( x_2 \):
\[ x_2 \left( \frac{M_2}{M_1} + 1 \right) = R \]
Solving for \( x_2 \):
\[ x_2 = \frac{R}{\frac{M_2}{M_1} + 1} \]
Now substituting \( x_2 \) back to find \( x_1 \):
\[ x_1 = \frac{M_2}{M_1} x_2 = \frac{M_2}{M_1} \cdot \frac{R}{\frac{M_2}{M_1} + 1} \]

### Step 6: Find the Ratio
Now, we can find the ratio of the distances traveled:
\[ \frac{x_1}{x_2} = \frac{\frac{M_2}{M_1} \cdot \frac{R}{\frac{M_2}{M_1} + 1}}{\frac{R}{\frac{M_2}{M_1} + 1}} = \frac{M_2}{M_1} \]

Thus, the ratio of the distances traveled by \( M_1 \) to the distance traveled by \( M_2 \) is:
\[ \frac{x_1}{x_2} = \frac{M_2}{M_1} \]

### Final Answer
The ratio of the distances traveled by \( M_1 \) to the distance traveled by \( M_2 \) is \( \frac{M_2}{M_1} \).

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